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Magazine Chapter 11: Problem 187
Find the specified term for the geometric sequence given. Let \(a_{1}=4, \quad a_{n}=-3 a_{n-1} .\) Find \(a_{8}\)
Short Answer
Expert verified
The 8th term is -8748.
Step by step solution
01
Understand the Problem Statement
We need to find the 8th term of a geometric sequence where the first term is given as \(a_1 = 4\) and each subsequent term is computed as \(a_n = -3 \times a_{n-1}\).
02
Calculate the Second Term
Using the recursive formula, the second term \(a_2\) is calculated as:\[a_2 = -3 \times a_1 = -3 \times 4 = -12\]
03
Find the Third Term
Using the recursive formula again, the third term \(a_3\) is:\[a_3 = -3 \times a_2 = -3 \times (-12) = 36\]
04
Calculate the Fourth Term
Continuing the sequence, the fourth term \(a_4\) is:\[a_4 = -3 \times a_3 = -3 \times 36 = -108\]
05
Compute the Fifth Term
Similarly, the fifth term \(a_5\) is:\[a_5 = -3 \times a_4 = -3 \times (-108) = 324\]
06
Determine the Sixth Term
The sixth term \(a_6\) is calculated as:\[a_6 = -3 \times a_5 = -3 \times 324 = -972\]
07
Derive the Seventh Term
The seventh term \(a_7\) comes out to be:\[a_7 = -3 \times a_6 = -3 \times (-972) = 2916\]
08
Calculate the Eighth Term
Finally, compute the eighth term \(a_8\):\[a_8 = -3 \times a_7 = -3 \times 2916 = -8748\]
09
Conclusion
The value of the eighth term \(a_8\) is found to be \(-8748\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Recursive Formula
A recursive formula is a fundamental way to express sequences, especially when one term depends on the previous. In this geometric sequence, the recursive formula is given by \( a_n = -3 \times a_{n-1} \). This means each term is calculated by multiplying the prior term by \(-3\).
To understand recursive formulas, remember:
To understand recursive formulas, remember:
- The formula always requires an initial starting point, like \( a_1 = 4 \).
- Each term in the sequence is derived from directly using the formula on the previous term.
Geometric Progression
Geometric progression, often called a geometric sequence, is a sequence where each term after the first is found by multiplying the previous term by a fixed number known as the common ratio. In this exercise, the common ratio is \(-3\).
In geometric progressions:
In geometric progressions:
- Consistency is key: Every term is generated using the same ratio.
- If the ratio is negative, like in our current situation, the sequence will alternate signs.
Sequence Calculation
Sequence calculation involves the step-by-step determination of each term based on the defined mathematical rule. Here, we use the recursive formula repeatedly to calculate each term leading to \( a_8 \).
To calculate the sequence:
To calculate the sequence:
- Start from the given first term, such as \( a_1 = 4 \).
- Apply the recursive formula to find the next few terms.
- Continue this process until reaching the desired term, ensuring each calculation follows the same multiplication rule.
